function?[TexpLexp]=new1lyapunov（nrhs_ext_fcnfcn_integratortstartstepttendystartioutpd）;
%
%????Lyapunov?exponent?calcullation?for?ODE-system.
%
%????The?alogrithm?employed?in?this?m-file?for?determining?Lyapunov
%????exponents?was?proposed?in
%
%?????????A.?Wolf?J.?B.?Swift?H.?L.?Swinney?and?J.?A.?Vastano
%????????“Determining?Lyapunov?Exponents?from?a?Time?Series“?Physica?D
%????????Vol.?16?pp.?285-317?1985.
%
%????For?integrating?ODE?system?can?be?used?any?MATLAB?ODE-suite?methods.?
%?This?function?is?a?part?of?MATDS?program?-?toolbox?for?dynamical?system?investigation
%????See:????http://www.math.rsu.ru/mexmat/kvm/matds/
%
%????Input?parameters:
%??????n?-?number?of?equation
%??????rhs_ext_fcn?-?handle?of?function?with?right?hand?side?of?extended?ODE-system.
%??????????????This?function?must?include?RHS?of?ODE-system?coupled?with?
%??????????????variational?equation?（n?items?of?linearized?systems?see?Example）.???????????????????
%??????fcn_integrator?-?handle?of?ODE?integrator?function?for?example:?@ode45??????????????????
%??????tstart?-?start?values?of?independent?value?（time?t）
%??????stept?-?step?on?t-variable?for?Gram-Schmidt?renormalization?procedure.
%??????tend?-?finish?value?of?time
%??????ystart?-?start?point?of?trajectory?of?ODE?system.
%??????ioutp?-?step?of?print?to?MATLAB?main?window.?ioutp==0?-?no?print?
%??????????????if?ioutp>0?then?each?ioutp-th?point?will?be?print.
%
%????Output?parameters:
%??????Texp?-?time?values
%??????Lexp?-?Lyapunov?exponents?to?each?time?value.
%
%????Users?have?to?write?their?own?ODE?functions?for?their?specified
%????systems?and?use?handle?of?this?function?as?rhs_ext_fcn?-?parameter.??????
%
%????Example.?Lorenz?system:
%???????????????dx/dt?=?sigma*（y?-?x）?????=?f1
%???????????????dy/dt?=?r*x?-?y?-?x*z?=?f2
%???????????????dz/dt?=?x*y?-?b*z?????=?f3
%
%????The?Jacobian?of?system:?
%????????|?-sigma??sigma??0?|
%????J?=?|???r-z????-1???-x?|
%????????|????y??????x???-b?|
%
%????Then?the?variational?equation?has?a?form:
%?
%????F?=?J*Y
%????where?Y?is?a?square?matrix?with?the?same?dimension?as?J.
%????Corresponding?m-file:
%????????function?f=lorenz_ext（tX）
%?????????SIGMA?=?10;?R?=?28;?BETA?=?8/3;
%?????????x=X（1）;?y=X（2）;?z=X（3）;
%
%?????????Y=?[X（4）?X（7）?X（10）;
%?????????????X（5）?X（8）?X（11）;
%?????????????X（6）?X（9）?X（12）];
%?????????f=zeros（91）;
%?????????f（1）=SIGMA*（y-x）;?f（2）=-x*z+R*x-y;?f（3）=x*y-BETA*z;
%
%?????????Jac=[-SIGMASIGMA0;?R-z-1-x;?y?x-BETA];
%??
%?????????f（4:12）=Jac*Y;
%
%????Run?Lyapunov?exponent?calculation:
%?????
%????[TRes]=lyapunov（3@lorenz_ext@ode4500.5200[0?1?0]10）;???
%???
%????See?files:?lorenz_ext?run_lyap.???
%??
%?--------------------------------------------------------------------
%?Copyright?（C）?2004?Govorukhin?V.N.
%?This?file?is?intended?for?use?with?MATLAB?and?was?produced?for?MATDS-program
%?http://www.math.rsu.ru/mexmat/kvm/matds/
%?lyapunov.m?i

?屬性????????????大小?????日期????時(shí)間???名稱
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?????文件???????4749??2005-10-22?22:55??new1lyapunov.m

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?????????????????4967????????????????????2